Page 69 - Mechanical Engineers' Handbook (Volume 4)

58 Fluid Mechanics

m . In differential form this is dV/V dA/A d / 0, which indicates that all three
3
quantities may not increase nor all decrease in the direction of flow. For incompressible duct
3
flow Q VA m /sec, where V and A are as above. When the velocity varies throughout a
cross section, the average velocity is
V udA
n
1
1
A n i 1 u i
where u is a velocity at a point, and u are point velocities measured at the centroid of n
i
equal areas. For example, if the velocity is u at a distance y from the wall of a pipe of radius
, u u (y/R) 1/7 and the average velocity is V 49 ⁄60 u .
R and the centerline velocity is u m m m

5 FLUID MOMENTUM
The momentum theorem states that the net external force acting on the fluid within a control
volume equals the time rate of change of momentum of the fluid plus the net rate of mo-
mentum flux or transport out of the control volume through its surface. This is one form of
the Reynolds transport theorem, which expresses the conservation laws of physics for fixed
mass systems to expressions for a control volume:

D
—
F V dV
Dt
material
volume
— V(V ds)

t V dV
control control
volume surface

5.1 The Momentum Theorem
For steady flow the first term on the right-hand side of the preceding equation is zero. Forces
include normal forces due to pressure and tangential forces due to viscous shear over the
surface S of the control volume, and body forces due to gravity and centrifugal effects, for
example. In scalar form the net force equals the total momentum flux leaving the control
volume minus the total momentum flux entering the control volume. In the x direction

F ( ˙mV ) ( ˙mV )
x entering S
x
x leaving S
or when the same fluid enters and leaves,
F ˙m(V x leaving S V x entering S )
x
with similar expressions for the y and z directions.
represents momentum flux passing a section and V is
For one-dimensional flow m˙V x x
the average velocity. If the velocity varies across a duct section, the true momentum flux is
(u dA)u, and the ratio of this value to that based upon average velocity is the momentum
A
correction factor ,